Why is that surprising? This sort of thing seems to be ubiquitous in our universe. F=ma is equivalent to the principle of least action. Turing machines are equivalent to the lambda calculus. Electrodynamics is equivalent to the existence of a finite reference velocity.
Also:
> You need to define temperature as an associative observable
Did you mean transitive? My understanding is that the required axiom is that if system A is in thermal equilibrium with system B, and B is in equilibrium with C, then A will be in equilibrium with C. That's transitivity, not associativity.
I corrected the wrong use of the word associative, thanks!
I submit: the qualification 'completely independent axioms' is incorrect. There is the observation: temperature is transitive. This transitive property is a statement of conservation (In terms of Carnot's thermodynamic it used to be thought of as conservation of Caloric.) The concept of Conservation of a quantity correlates with information.
We have that statistical mechanics subsumed Carnot's thermodynamics.
The laws of Carnot's thermodynamics are theorems of statistical mechanics. (Those theorems weren't necessarily stated explicitly. I'm saying the principles of statistical mechanics are sufficient to imply the laws of Carnot's thermodynamics.)
However, it is extremely valuable to also understand that the vast majority of thermodynamics (including our understanding of entropy) can be derived from completely independent axioms that have no relationship to information theory or the knowledge of discreteness of matter. You need to define temperature as an associative observable (zeroth law of thermodynamics), you need to define heat as a type of energy (first law of thermodynamics), and you need some version of the second law (there are a few equivalent ones, but the easiest is that "heat flows only from high temperature to low temperature"). Lastly, you need equations of state as a starting point (instead of assumptions about the microscoping constituents of matter). Tadaaa, you have all that is necessary to derive all our knowledge of thermodynamics without knowing statistics or atoms.
Aesthetically this is incredibly pleasing and intellectually it is quite surprising that these two sets of axioms that look nothing alike produce the same self-consistent physics. To be productive in cutting edge research you usually need to be comfortable with both.